The reaction for which a nuclear reactor is named is the atoms of unstable substances rupturing on a subatomic level.

Every substance is made of atoms.

Atoms that share the same number of protons in their nucleus are the same element. The protons are all 'positively' charged and want to repel each other and fly apart, but they cannot because neutrons got them stuck together. The combined positive charge of the neutrons, though, attracts and captures electrons (which are negatively charged) in their orbit.

Sidebar: it is the interactions between the electron shells of atoms that allow atoms to stick together to form molecules. For instance, water is one hydrogen atom and two oxygen atoms.

Atoms with one proton in the middle are hydrogen. Atoms with two protons are helium. Atoms with three are lithium, beryllium has 4, boron has 5, carbon has 6, atoms with seven protons are nitrogen, atoms with eight protons are oxygen. And so on. The entire list of all known atoms is the periodic table of elements, and the atomic number of each element is how many protons it has in its nucleus.

Another sidebar: atoms can sometimes have an extra electron, or be missing an electron. These are "negative" and "positive" ions. Lithium ion batteries, for instance, operate on a principle of chemical reactions that can store extra electrons when charged, and strip those electrons off and release them when discharged.

Less of a sidebar because this bit is getting relevant to nuclear/atomic energy: atoms can have a varying number of neutrons too. Hydrogen only has one proton so it doesn't even necessarily NEED a neutron. If it has a neutron, it is significantly heavier than a hydrogen atom that doesn't have a neutron, and we call it deuterium. It can even have TWO neutrons, and be nearly three times as heavy as a result of the extra particle, and we call it tritium. the varying numbers of neutrons in an atom's nucleus are isotopes of an element.

Recap:

• An elemental unit of matter is an atom and it is almost always made of protons, neutrons, and electrons.
• What that matter "is" and what that matter "does" is determined by the number of protons.
• Protons are positively charged, electrons are negatively charged, and neutrons have no charge.
• Neutrons bind protons together at the nucleus so their positive charge doesn't make them fly apart.
• The number of electrons orbiting the nucleus can vary, and when it's not equal to the charge of the protons, the atom has been "ionized" and is called an "ion" of that element.
• • if there are extra electrons, it's a negative ion; and if there is a deficit of electrons, it is a positive ion.
• The number of neutrons inside the nucleus can vary, and each neutron has a significant mass, comparable to the mass of the protons.
• • The total number of particles (neutrons plus protons) in the nucleus of an atom has a significant influence on the mass of the atom.
• • We call the different counts of total nucleus particles for the same number of protons "isotopes".

Now I can finally tell you what nuclear fission and nuclear fusion are about.

Fusion is when atoms (usually very light ones) under titanic, gargantuan, nigh incomprehensible pressure are forced together so close, under so much force that it overcomes the negative-to-negative electrostatic repulsion of their electron shells, that the nuclei of the atoms get close enough that they suddenly stick together, merging their assemblages of neutrons and protons into a single nucleus and the electrons all sharing that orbit.

Very light atoms such as hydrogen and helium can have an easier time fusing if there are more neutrons present in their nuclei, assisting with the 'stickiness' (not a technical term) of each atom's nucleus to stick to each other. When we do fusion here on earth, we can't achieve the pressures necessary for regular hydrogen or helium to fuse, so we use deuterium or tritium to do it instead.

Meanwhile, Fission is when atoms (usually very heavy ones with lots of extra neutrons) break apart. Isotopes of very heavy elements with abnormally high numbers of neutrons behave differently from their more stablely balanced 'not too many neutrons' related isotopes. The nucleus can become 'unstable' and prone to breaking. You could imagine this, metaphorically speaking, as a physics engine that's having to deal with too many rigidbody collisions between too many objects in a tight space, with the objects clipping into each other and building up incredible amounts of un-accounted-for forces which, when crossing an escape threshold, cause the pile of objects to break apart.

If you have a relatively stable isotope that will become a very UNSTABLE one if you just add another neutron, then you can cause it to break apart (fission) by shooting a neutron at it. And actually hitting. Now, if you have a whole crapton of these relatively stable atomic isotopes collected together (refined into nuclear fuel), you can shoot a neutron at that blob of atoms and statistically ONE of them is gonna get hit with that neutron and break apart.

When an atom breaks apart, it basically explodes very fast and that's a lot of kinetic energy. Kinetic energy on an atomic level, well, it hits other atoms which hit other atoms and they all vibrate and that's what we call heat.

But that's not all. When the atom breaks, it will release extra neutrons that it can no longer hold onto. IF it releases more than 2.1 neutrons on average when it breaks, those two neutrons will go flying off and statistically at least one of them will hit another atom of the same substance, the same isotope, with the same 'just on the cusp of blowing apart' situation, causing IT to fission too, and ALSO shoot off a few neutrons. Those also hit barely stable atoms that become unstable and fission releasing neutrons which then destabilize other atoms which fission and shoot off neutrons which then fission other atoms that fission other atoms... This is called criticality and it's the tipping point at which a nuclear fission reaction can sustain itself.

In order to sustain this reaction, we build a structure that we put the fissile fuel into, a structure specifically designed--with specific materials specifically shaped--to reflect the neutrons back into the fuel so that the reaction can keep going. This is a nuclear reactor core. By inserting substances, meanwhile, that will absorb neutrons and slow the effect down OR by withdrawing the fuel rods from the 'sweet spot' in the reactor core, we can control the intensity of the reaction so it doesn't blow up EVEN BIGGER, and therefore we call these Control Rods.

And that's the essential fissile chain-reaction that is core to the operation of a nuclear power plant. Every single one of those fissioning atoms releases a bunch of heat and that heat adds up. A thermal transfer fluid of some kind surrounding the core will absorb allllll that heat, and carry it to a heat exchanger that dumps all that heat into yet another working fluid, this one whose job is to boil FURIOUSLY when it gets hot enough and generate a crapton of vapor pressure, which then is allowed to blow through and thereby push turbines.

That's fission nuclear reactor power!

The fuel becomes hot because the nuclear reaction in it is producing both light (eg: gamma rays) and fast-moving subatomic particles. These both interact with the rest of the fuel to heat it up.

It's natural radioactive decay producing heat except it is artificially concentrated and accelerated.

If left in nature they would not produce anything near the heat they produce when we mess with them

The atoms of plutonium or whatever break down into other atoms. This process is called fission. When it breaks down it also lets neutrons loose, which then attach to other plutonium atome, destabilizing it, then the next one and so on in a chain reaction that produces heat. That heat is transferred to water, which the eventually powers steam turbines. Electricity does not come directly from the rods.

This whole process is regulated via neutrons(or maybe protons, not 100% on that). Put the rods of plutonium closer together, get more fission, more heat. I think they use graphene to regulate things too.

It's basically the former: the radiation given off by the atoms breaking apart triggers more fission when it bumps into other unstable atoms, which just keeps compounding if you have a bunch of fuel in one place until it gets REALLY hot and melts.
To prevent the meltdown you put control rods throughout the fuel which absorb the excess radiation and keep the fuel at useful temperature, roughly like how a fuse is built to burn at a consistent speed rather than simply exploding.